Russell’s Paradox

Let «A» be a set and «x» is an element of «A» if and only if «x» is not an element of «x». In other words: «A is a set of all sets that don’t contain themselves as a members».

Why is this a paradox?. Well, imagine that there are several lists of things, let’s take a list named «x», now, you’ll include the list «x» in «A» if and only if «x» doesn’t have «x» in the list. Does «A» belong to the set «A»?.

Suppose that «A» doesn’t belong to «A», then by definition, «A» belongs to «A». This is contradictory.

Now suppose another case, «A» belongs to «A», then, «A» just can’t belongs to «A» by definition of «A» set. Contradiction.

Another way to see this paradox is a classical situation:

In a town there is just one male barber, and he shaves to every man who don’t shaves themselve. And then, we got two situations:

If the barber doesn’t shaves himself, then he shaves himself.

If the barber shaves himself, then by definition, he doesn’t shave himself.